Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

TL;DR

This paper explores the convergence properties of multipoint Schur algorithms and orthogonal rational functions, developing a rational analogue to Szegő theory for cases with accumulation points on the unit circle.

Contribution

It introduces a new rational Szegő theory and generalizes existing results to cases with accumulation points, advancing the understanding of convergence in multipoint Schur analysis.

Findings

Convergence of Wall rational functions established.

Development of a rational Szegő theory for accumulation points.

Asymptotic behaviors of orthogonal rational functions analyzed.

Abstract

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.